Pair distribution function definition

Generalize this definition to dots in three dimensions and verify that it is the pair distribution function of statistical mechanics. 

The pair distribution function leads to the pair correlation function, which illustrates how the local order found near a given molecule is lost as distance from the molecule increases. This quantity is of fundamental theoretical interest and may be determined in X-ray and neutron scattering experiments, as discussed below. The definition of the pair correlation function g r j) is... 

Comparing the inner integral with the definition of the pair distribution function given by equations (2.4.4) and (2.4.7), one may now write... 

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

Clearly, due to the equivalence of the particles, we could have chosen any other two indices. The third and fourth steps make use of the definition of the pair distribution function defined in section 2.2. 

We start by considering the grand canonical ensemble characterized by the variables T, V, and fi where p = (/q, p2,..., pc) is the vector comprising the chemical potentials of all the c components of the system. The normalization conditions for the singlet and the pair distribution functions follow directly from their definitions. Here, we use the indices a and [l to denote the species a, jS = 1, 2,..., c. The two normalization conditions are (for particles not necessarily spherical)... 

This can be obtained directly from the definition of the pair distribution function (section 2.2) by putting UN= 0 for all configurations of the N particles. 

By exchange, we mean the density functional definition of exchange, in which the wavefunction is a Slater determinant whose density is the exact density of the interacting system, and which minimizes the energy of the non-interacting system in the Kohn-Sham external potential, iv,a=o- Another useful concept is the pair distribution function, defined as[14]... 

In organic polymers, there is always more than one kind of atoms present. Even a hydrocarbon polymer, the simplest among polymers in chemical structure, is made up of two types of atoms, carbons and hydrogens. To specify the short-range order in a hydrocarbon polymer, three partial pair distribution functions, g cc(r), gHH(r)> and ch(t)> are needed. The definition of gch(/0, for example, is as follows. We pick an arbitrary C atom and look at the volume element dr at a position displaced from it by r. We note the number of H atoms present in this volume element to be equal, on the average, to ncw(r) dr. The function gcn(r) is obtained on normalization as... 

The notions of molecular distribution functions (MDF) command a central role in the theory of fluids. Of foremost importance among these are the singlet and the pair distribution functions. This chapter is mainly devoted to describing and surveying the fundamental features of these two functions. At the end of the chapter, we briefly mention the general definitions of higher-order MDF s. These are rarely incorporated into actual applications, since very little is known about their properties. 

The normalization conditions for the singlet and the pair distribution functions follow directly from their definitions. Here, we use the indices a... 

The idea behind this definition is that the more hydrogen bonds in the system, the more structured it is. The factor J has been included in (6.113) to avoid counting each HB twice. Relation (6.113) can be transformed into an equivalent form, using the pair distribution function,... 

Since all of these terms are equivalent, we simply choose one specific term and multiply it by M. This gives the expression which is identical to the definition of the pair distribution function, Eq. (4.3.36). 

Attempts to improve the theory by solving the Poisson-Boltzmann equation present other difficulties first pointed out by Onsager (1933) one consequence of this is that the pair distribution functions g (r) and g (r) calculated for unsymmetrically charged electrolytes (e.g., LaCl or CaCl2) are not equal as they should be from their definitions. Recently Outhwaite (1975) and others have devised modifications to the Poisson-Boltzmann equation which make the equations self-consistent and more accurate, but the labor involved in solving them and their restriction to the primitive model electrolyte are drawbacks to the formulation of a comprehensive theory along these lines. The Poisson-Boltzmann equation, however, has found wide applicability in the theory of polyelectrolytes, colloids, and the electrical double-layer. Mou (1981) has derived a Debye-Huckel-like theory for a system of ions and point dipoles the results are similar but for the presence of a... 

A third form of the basic scattering equation is obtained if structures are characterized with the aid of the pair distribution function (r). Per definition, the product... 

We still need an expression for the cross-response coefficients. The Debye-structure functions 5d(-Ra ) and 5D(i B ) are the Fourier-transforms of the pair distribution functions QAAi f ) and gBBi f") for the A- and B-monomers within their blocks. Considering this definition, it is clear how the coefficients for the cross responses and should be calculated. Obviously, they must correspond to the Fourier-transforms of the pair distribution functions gAB r) and gBAi f ) which describe the probability of finding a B- or A-monomer at a distance r from a A- or B-monomer respectively. Actually, both are identical... 

As follows directly from the definitions, the density distribution and the pair distribution function are related by... 

We define an "i-th nearest neighbour complex to be a pair of oppositely charged defects on lattice sites which are i-th nearest neighbours, such that neither of the defects has another defect of opposite charge at the i-th nearest neighbour distance, Rit or closer. This corresponds to what is called the unlike partners only definition. A different definition is that the defects be Rt apart and that neither of them has another defect of either charge at a distance less than or equal to R. This is the like and unlike partners definition. For ionic defects the difference is small at the lowest concentrations the definition to be used depends to some extent on the problem at hand. We shall consider only the first definition. It is required to find the concentration of such complexes in terms of the defect distribution functions. It should be clear that what is required is merely a particular case of the specialized distribution functions of Section IV-D and that the answer involves pair, triplet, and higher correlation functions. In fact this is not the procedure usually employed, as we shall now see. 

We follow here the notation of Thakkar and Smith [15], and evaluated this and the other distribution functions mentioned below using the formulas given by them. The prefactor 2 in the definition of D(ri) causes it to describe the pair density contributions of the entire electron distribution (rather than that of one of the two electrons). 

Using the pair-wise additivity of U(R), it is possible to integrate Eq. (18) over the equilibrium configurations of (N — 2) particles. If one then uses the definition of the radial distribution function, an expression for E in terms of g( r) and u(r) is obtained, and it is referred to as the energy equation... 

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